key: cord-0597095-pepgd70t
authors: Ferreyra, Emanuel Javier; Jonckheere, Matthieu; Pinasco, Juan Pablo
title: SIR dynamics with Vaccination in a large Configuration Model
date: 2019-12-27
journal: nan
DOI: nan
sha: 5c77923e37d0e5ca28ed67a78fd758c29bf3cf12
doc_id: 597095
cord_uid: pepgd70t

We consider an epidemic SIR model with vaccination strategy (both centralized and individual) on a sparse configuration model random graph. We show convergence and characterized the scaling limits of the system when the number of odes grows. Then, we study and characterize the optimal controls for the limiting equations formulated as in the framework of game theory both in the centralized and decentralized setting. Finally, we show how the characteristics of the graphs influence the vaccination efficiency for optimal strategies.

Since the study of vaccination cannot be easily considered in vivo, there is a long history of quantitative analyses of epidemics and policies to avoid them, through mathematical models and (more recently) their simulations. Besides, very similar dynamics (epidemics and vaccination) could also be used to model the spread of negative information (like rumors or fake news) from which a social-media user could be protected (vaccinated) using efficient facts verification. In both cases, the use of simulations and mathematical and models have proved to be instrumental to evaluate vaccinations strategies and epidemics characteristics. We refer the interested reader to the complete and pedagogical review [1] which contains both historical and modern references on the epidemic-like processes on complex networks.

In this work, we aim at describing optimal strategies of vaccinations and their performance, modelling the interactions between individuals as a sparse random graph with a given degree distribution. Our motivation steams from the fact that in recent years, many authors pointed out the non-lattice characteristics of interactions networks (both in social media and physical social interactions), as well as the presence of large degree nodes or hubs which had a strong impact on epidemics characteristics and their vaccination counterparts. As a consequence, results for lattice-like graphs and for mean field models (i.e. complete graphs) were shown to have a quite limited modelling power.

On the other hand, while SIR epidemics with or without vaccination have been studied extensively in the context of mean-field models where each individual potentially interact with every other individual [2] , there has been more recently a research effort to include the effect of local interactions or neighborhoods by modeling the potential interactions between individuals as a sparse random graph [3] , [4] . In this context, the effect of non-homogeneity in the inter-individual interactions and the famous phenomena of small word can be reflected in the model by defining a degree-distribution describing the statistics of the interactions.

In [5] , a set of finite-dimensional differential equations describing the asymptotic of a SIR dynamics on a sparse Configuration Model (CM) has been rigorously derived as projections of an infinite dimensional system describing the scaling limits of the degree measures (between susceptible, infected and recuperated). This is a remarkable result as it allows to grasp both the specifics of the interaction graph and the epidemic dynamics in a simple 5-dimensional deterministic dynamic system. Moreover, it agrees with other representations from the physics literature including the work of Volz which described a Poissonian SIR epidemics using coupled non-linear ordinary differential equations. The results of [5] hence show that the latter equations are indeed verified in the thermodynamic limit (i.e. , when the number of nodes tends to infinity in a sparse configuration model). See also [6] for more on the SIR dynamics on the configuration model and [7] for equivalent formulation of the ODE dynamics.

Contribution. In the present work, our starting point consists in generalizing the results of [5] for SIR on CM by incorporating a strategy of vaccination depending (in a generic manner) on time and depending linearly on the node degree (i.e., the number of potential individuals with which one can interact). Using scaling limit techniques (following [5] and [6] ), we show the convergence of the degree measures in this more general context and obtain a generic description of the epidemic dynamics subject to a vaccination strategy. As in the case without vaccination, we are able to derive a finite-dimensional differential system describing the evolution of the quantities of interest (as the number of infected, vaccinated, recovered,...).

We then study the optimal controls for the vaccination formulated as a game both in a decentralized and centralized setting. The decentralized case is approached from the theory of mean field games, regarding the perspective of a single rational individual added to an infinite population with an arbitrary vaccination policy. We show in particular that the optimal vaccination strategy boils down to a bang-bang control, i.e., the optimal solution consists in vaccinating with the highest possible rate until some fixed time-threshold depending on the connectivity of the network and the costs, and then not to vaccinate at all anymore. On the other hand, using techniques from continuous optimization for systems with restricted controls, we also show that the optimal centralized strategy is of a threshold type The paper is organized as follows. In Section §2 we introduce the model and the notation. In Section §3 we present the main result of this work. Section §4 is devoted to the optimal control problems, and we show that the vaccination strategy is threshold. In Section §5 we conclude by analyzing the effect of the vaccination policy on the epidemics. The full proofs of the different results can be found in Section §6.

Before exposing the dynamics, we introduce briefly the stochastic environment of the epidemic process. We use the Configuration Model random graph introduced by Bollobás [8] which can be constructed as follows. Suppose we have N nodes, and a sequence of degrees d 1 , . . . , d N independent and identically distributed according to (p k ) k∈N 1 . We first assign a quantity d k of half-edges to each node k and then choose two of them uniformly from the unmatched ones, establishing the connection between the nodes, until all the half-edges are matched. This procedure may lead to a multi-graph, but the probability of having a simple one is bounded from below independently of N by a strictly positive constant if we assume finite second moment, so we may repeat the algorithm until we obtain a simple graph [9] .

As a consequence the degree of a randomly chosen node x is distributed according to p k , and if we consider an edge (x, y), the probability that y has degree k is kp k j∈N jp j , which is so-called the size-biased degree distribution.

We now describe the dynamics of the epidemics with vaccination. For a given Susceptible node (i.e. not having contracted the illness nor vaccinated) we consider several exponential clocks with parameter r, one for each edge with an Infected alter (i.e., potential encounters). It will describe the contact process: if this clock rings, the Susceptible makes a transition to state Infected and remain infectious during an exponential time with mean 1/β, whereupon it will not longer infect any node, going to Recovered state. Also for the Susceptible, we consider another exponential clock with parameter π t (k) depending on the degree k of the node, which is the time dependent control variable and represents the rate at which the individual becomes Vaccinated. Both Recovered and Vaccinated are absorbing states of the resulting continuous time Markov chain, and we differentiate between both to keep track of the epidemics.

We denote S t , I t , R t and V t the total number of Susceptible, Infected, Recovered and Vaccinated nodes respectively, which are important in the description of the infectionvaccination process. These quantities are of central interest in most of the epidemics literature and are the main variables on which equations of the dynamics are exhibited (we refer the reader to [2] for an informative review on epidemics dynamics). Nevertheless, let us remark that our model being over a configuration model random graph, computing its dynamics is in principle very demanding, as we should study a stochastic process in (growing) dimension N, the number of nodes.

Instead, we are able to study the dynamics of four measures describing the connection between the susceptible population and the rest. For this purpose, we resort to the so-called principle of deferred decisions, revealing the graph simultaneously with the propagation of the disease, regarding the types of the edges connecting the different states of the individuals. This trick is possible since the random environment for the epidemics dynamics act over a configuration model (which is constructed using a uniform matching). We follow here the ideas developed in [5] and [10] .

We now describe these dynamics precisely. For a given vertex x, we denote d x (S) the random number of edges (x, y) with y a susceptible individual. The measure µ S t (k) denotes the number of susceptible nodes with degree k at time t,

Similarly,

represent the number of nodes in each state I, R, V connected with the susceptible population. We write µ t = (µ S t , µ IS t , µ RS t , µ V S t ). Another crucial parameter of the model is the probability that a node of degree one remains susceptible until time t, which we denote

Where p I t is the probability that an edge from a susceptible individual to link to an infectious neighbor and pi t is the vaccination rate, at time t.

We show that the degree empirical measures of each type converge to an infinite system of differential equations as the population size tends to infinity. The corresponding deterministic solution will in turn give interesting insights on the effect of the vaccination in the propagation of the epidemic for large populations. i) There exists a unique solution µ t of the deterministic system of differential equations (2) . ii) When n goes to infinity, the sequence (µ (n) ) n∈N converges in distribution to µ in the Skorokhod space.

The proof is very similar to the proof of the main theorem of [5] but we add it in the last section for a matter of completeness. Choosing f (k) = 1 i (k) we obtain a countable system of ordinary differential equations that allows us to describe the infection propagation in terms of the measures.

In most of the literature, the object of study is the proportion of the population in each state, explaining the infection numbers. Our work describe the dynamic of the following edge-based quantities

the number of semi-edges connecting a susceptible node; and N IS We also consider the proportion of edges associated to those quantities:

It is a key point in our model to fix how much information the individuals have in the decision making process. In the sequel we suppose that the vaccination strategy of each individual is proportional to her degree with the same proportionality constant for each agent in the population, i.e., π t (k) = π t k.

As in the case without vaccinations we use the generating function for the initial degree distribution g(x) = k∈N µ S 0 (k)x k in order to reduce the number of dimensions (from infinity) to six, allowing in this way a useful description of both the epidemics and the optimal vaccination strategy. Proposition 1. The quantities x = (α, I, V, p S , p I , p V ) defined by the measure µ satisfy the following system of differential equations of the formẋ = ϕ(x, π),

Moreover, ϕ is Lipschitz in the variable x and uniformly bounded, and therefore by [11] , the problem (3) admits a unique solution for each initial datum and a measurable π :

Proof. We use the generating function to compute a closed expression for N S t , N IS t , N RS t and N V S t and its derivatives. We denote 1(x) := 1 and

is the number of susceptible individuals at time t.

In a similar way,

The next step is to find the dynamics for p S t and the other edges probabilities. Before calculating it, let us note that:

Using the definition of p I

We replace f by χ in (2), and after some basic calculation and rearranging of terms by using the multinomial theorem, we havė

Reasoning in much the same way with the other probabilities, and putting all the equations together, we have, for each control π : [0, T ] → [0, ν], a closed system of equations that generalizes the one proposed by Volz:

We can see the strong dependence on the initial distribution and its size biased distribution, hidden in the expression αtg (αt) g (αt) , the expected number of susceptible individuals connected with a neighbour of a given degree at time t.

Regarding the results on asymptotic similarity between a configuration model with Poisson degree distribution and the classical Erdös-Renyi random graph [12] we relate our model with the Mean Field point of view. In the sparse Erdös-Renyi model, the number of neighbours in the graph follows a binomial distribution, which can be approximated in a large population by a Poisson distribution. On the other hand, when the graph is fully connected and the contact process is determined by a Poisson process, the number of neighbors with whom each node effectively connects is also distributed Poisson.

In the Mean Field model [13] , an individual of an homogeneous population encounters others following a Markov process in continuous time with rate r. Slightly differently from our model, individuals can be in the four states: susceptible, infected, recovered or vaccinated; and we denote S t , I t and R t , its respective proportions of the total population. Here, vaccinated individuals are treated as recovered, since the influence in the propagation of the disease is the same. If the initial individual of the contact process is susceptible and the encountered one is infected, the first become infected. An infected node recovers at rate β, and a susceptible can choose its own vaccination rate π, going to recovery state. When the size of the population goes to infinity, the dynamics of the population where all players use the vaccination strategy π is described by the following system of equations:

The main difference between the two systems lies on the term associated to the infection process in the mean field case, rS t I t , is now rp I t α t g (α t ) = rN IS t and the neighbors are chosen according to the size biased distribution. In the particular case of Poisson distribution, the size biased distribution is also Poisson, and as we show in this section, both models are asymptotically similar.

The model is different because we are regarding a local dependence on the interactions, looking at an edge-based dynamics instead of an individual-based one. In the propagation of the disease, exponential clocks of the contact process are assigned to edges connecting susceptible with infected, the mechanism is not to choose uniformly between all the population but in a neighbourhood, this modifies qualitatively the generator of the MDP and therefore the limit equation.

In Figures 1 and 2 we present the long time evolution of the proportions of susceptible, infected, vaccinated and recovered nodes for the Mean Field Model (left panel) and the Configuration Model (right panel), for different parameters and a xed vaccination strategy. We take an ε-proportion of initially infected, this give us: We can see that the simulations are very close, though not exactly similar. We can actually give an analytical argument to explain this similarity. Let us consider a population in which every individual has C possible contacts and scale it such thatr = rC remains constant. The Mean Field equation we get before the scaling is the same as (7) replacing r and π byr andπ respectively.

We prove that our dynamics on a configuration model with Poisson degree distribution, not fully connected but uniformly linked, also solve these equations while C goes to infinity (and therefore N → ∞). Let us consider the case g(x) = Ce C(x−1) , we know S = g(α), and

Sincer is taken to be constant, it is of order O(1) as C grows, and only a proportion of order O(I/C) of the edges may transmit the infection from one infected neighbor to the observed susceptible. With this, for large C, p I can be approximated by I − O(I/C) and similarly, when C is large enough, α is 1 − O(1/C). Moreover, p I α = I + O(I/C), giving uṡ

which is asymptotically the first equation in (7). The third follows from a similar reasoning, and the second from the fact that S = 1 − I − R.

The optimal vaccination problem has been studied in recent years from the perspective of control theory and game theory. There are two main points of view to analyze the population behaviour: as a rational individual immersed looking for its own benefit, or as a centralized agent who takes decisions for the overall, for example the government. This benefit can be thought of as to minimize the costs of a vaccination program or to prevent the epidemic spread of a disease. In [14] , the authors analyze the vaccination at the beginning of the period of time in consideration proposing an evolutionary game-theoretic problem, where individuals use evidence to estimate costs of vaccination, and the model is based on the agent point of view. In [13] , the authors apply mean field game theory to define and analyze the existence of an equilibrium in an infinite homogeneous population.

We consider here a vaccination strategy as a bounded and measurable time dependent function, followed uniformly by all the population but depending linearly on the connectivity of an individual. This is already a large family of controls, needing a quite general theoretical treatment, involving in particular weak and viscosity solutions.

In this section, we analyze the optimal control problem from an individual point of view. We focus on the perspective of a particular individual immersed in the population, who will take the optimal decision in order to minimize her cost regarding the behaviour of the population corresponding to a global vaccination policy.

This rational individual can be considered as a player in a game against the whole population with a fixed strategy, and we are therefore in the context of Mean Field Game theory, from which we will use the definitions of equilibrium and existence results. The scenario will be described by the limit system of equations we get in the previous section, and we suppose an individual is added to the population which evolves according to the vaccination strategy π. Since the population is infinite, the behavior of this new individual will not affect the evolution of the whole population, hence its dynamics will be described by the already stated equations (3), in the formẋ = f (x, π). We denote byπ the vaccination strategy for the new individual and (S t ,Ĩ t ,R t ,Ṽ t ) her probability distribution over the four possible states.

We suppose that this new individual has a degree distributed as the initial susceptible population µ S 0 , therefore, we have thatS t = g(α t ) whereα t = exp(− t 0 rp I s + π 0 s ds) and her state evolution will be determined by the following system of the formẋ = f 0 (x,x, π,π):

The new individual wants to minimize her cost defined by:

So, the new individual looks at the best response to strategy π, this is, she wants to play BR(π) ∈ arg minπC t (π,π). The minimum is taken over

which is a compact set for the weak topology. This implies that BR(π) is not empty, since any minimizing sequence has a limit. We say that π M F E is a mean-field equilibrium if and only if it is a fixed point of the best response functional, this is, π M F E ∈ BR(π M F E ).

The existence of this equilibrium follows from Theorem 2 in [15] , since the cost is linear in I, the function g is analytic, and the rates of transition for the new individual depend linearly in p I or are constant for a fixed π.

Although this result guarantees the existence of equilibrium, we will compute the solution analyzing our problem as a continuous time Markov Decision Problem with finite horizon.

Denote J S (t), J I (t) the optimal cost starting at time t in states susceptible and infected, respectively. The optimal cost J and the strategyπ * that realizes it, satisfy the the following Hamilton-Jacobi-Bellman optimality equation [16] :

We can see from the third line in (11) that

Hence J I decrease from J I (0) = C I β (1 − e −βT ) to J I (T ) = 0. We also realize that if J S (t) > c V thenπ(t) = 0. Since J S (T ) = 0 and the costs are continuous,J S (T ) = 0 therefore, if we call θ the first instant at which J S is below c V , we have J S (t) ≤ J I (t) for all θ ≤ t ≤ T , such that the second term in the second equation in (11) is non-negative. If J S does not cross c V then we take θ = 0.

Moreover, if J S > c V , the derivativeJ S is lower than itself when the condition J S ≤ c V holds, hence J S (t) ≤ J I (t) for all 0 ≤ t ≤ T and therefore J S is always decreasing before θ.

We have proved thatπ

Now we consider the total population as the optimizing agent. We first consider the control system of the formẋ = ϕ(x, π) like (3) where the set Π of admissible controls is compact, and the family of admissible control functions π is only restricted by its measurability. Given the initial data x(0) = x 0 the Cauchy problem has a unique solution, as we stated in Proposition 1.

Given an initial data (s, y) we consider the general optimization problem:

minimize : J(s, y, π) = T s L(x(t), π(t))dt + Ψ(x(T ))

where x depends not only on time but on the control and the initial data and the minimum is taken over Π the set of measurable functions π : [0, T ] → [0, ν].

As stated by the dynamic programming method, the optimal control can be characterized by the value function V (s, y) := inf π∈Π J(s, y, π), but the classical point of view does not allows discontinuous control functions. Hence we are now verifying the hypothesis that our setting must hold in order to have existence and uniqueness of the optimal control, based on more general results on viscosity solutions theory [17] , [18] . In the case of measurable control we can also apply the Pontryagin's Maximum Principle with less restrictive assumptions.

According to Lemma 9.2 in [17] , the functionals involved must satisfy

for all x 1 , x 2 ∈ R 6 , and π ∈ Π, for some constant C. Under this assumptions the value V is a bounded, Lipschitz continuous function, and it can be characterized as the unique viscosity solution to a Hamilton-Jacobi equation.

As a particular case inspired in the individual optimization problem exposed above, we define

We can easily check that our setting holds (14), by basic calculations and bounding the second term using the regularity of g and the Mean Value Theorem.

Further, given the data x(0) = x 0 , let t → x * (t) = x(t, π * ) be an optimal trajectory corresponding to the optimal control π * . Following Theorems 7.18 and 11.27 in [18] , there exists an absolutely continuous application t → p(t) ∈ R 6 called the adjoint vector, and a real number p 0 ≤ 0, such that (p, p 0 ) is non trivial, and such that for almost every t ∈ [0, T ]

where the Hamiltonian of the system is H = p 0 L 1 + pf .

Proposition 3. Let π * the strategy that minimizes (13) . Then π * is threshold.

Proof. Writing the equation for π * we get

which is a quadratic function of π with negative principal coefficient, and whose roots are 0 and ρ * . Since we are minimizing over π ∈ [0, ν] we can conclude that

and the proof is finished.

Since it is impossible to solve analytically the system (16), we apply the method of Forward-Backward Sweep presented in [19] in order to understand the behaviour of ρ * . We can see from the simulation that ρ * is negative at the beginning and monotone increasing.

As in the preceding section, optimal vaccination policies is of bang-bang type, indicating that vaccination must be intended with the maximum effort (maximum rate) and otherwise not to vaccinate.

When we consider the role of a government actor, this kind of problems can explain how vaccination policies, given a cost and a maximum vaccination rate, can interfere in effective immunization of the population, avoiding the epidemics at a mild economical cost. The budget in question may be modelled through a bound on the vaccination rate, and the individual behaviour contrasted with the social regime may justify the necessity of subsidies.

Knowing the strength of the disease in consideration and its recovering rate, the vaccination budget represented in ν, and taking in account the connectivity of the population, our work contributes to find this optimal vaccination effort.

We can calculate, as in previous works on epidemics over social networks, when does the epidemics occur in terms of the connectivity of the graph. This is deduced from the equation

taking the initial conditions previously stated in (8) we arrive to the already know threshold:

.

Here we cannot observe the influence of the vaccination, since at time zero this process has not started yet. Some of the previous research consider a previous immunization step [14] . Our work focus on a more realistic scenario where vaccination occur at the same time scale than infection. Also our work would generalize to vaccination at time 0, when the maximal vaccination rate diverge. In that case, the threshold strategy would not converge to a Dirac mass in 0. Following Miller [7] , we can compute the final epidemic size in our model. We compute ξ k , the probability that a randomly chosen susceptible node with degree k is never infected, considering the probabilities of infection, recovering and not vaccination at time T :

If we assume a bang-bang control π t = ν1 [0,θ] (t), after some simple calculations we get

Now, we can also compute the probability that an initially susceptible node remains susceptible after the epidemic: ξ = ∈N kπ k µ 0 (k) = r r + β g(e −θν ).

We can similarly compute the total number of recovered agents, from where analyze the spread of the disease. First we compute the final number of vaccinated agents,

Hence, V (∞) = S 0 − g(e −θν ) and therefore

Here, we can see the strong dependence of the connectivity of the network and the maximum rate of vaccination in order to reduce, exponentially, the propagation of the epidemic. Nevertheless the network characteristics do appear only through the generating function g. Additionally, the maximum rate of vaccination can be translated in the budget of the decision maker, because it may indicate how effective may be the decision to vaccinate.

Inspired in [5] and [20] we will represent the behavior of our dynamic as a process which is solution of a system of a stochastic differential equations derived from Poisson point measures (PPM).

We will use three different PPM for each event which modifies the quantities we are interested in: an infection, a recovery, or a vaccination. We need to identify the rates of this events and how to update the measures on the graph.

Suppose an event occur at time T , and let us analyze the first case, an infection. For that, it is convenient first to consider

the rate of infection of a given k-degree individual at time T . She will have her halfedges connected according to the quantities µ T and distributed following a multivariate hypergeometric distribution. We denote

Finally, given k, j, l and m, we have to update the measures µ IS T , µ RS T and µ V S T choosing the infected, recovered and vaccinated individuals who will be connected to the newly infected. In order to do that, we draw three vectors u = (u 1 , ..., u I T − ), v = (v 1 , ..., v R T − ), and w = (w 1 , ..., w V T − ) indicating how many links each I, R or V node has with the newly infected. We consider U = n∈N (N 0 ) n and for each µ ∈ M F (N 0 ) and n ∈ N we define U ⊇ U(µ, n) := u = (u 1 , ..., u µ,1 ) :

and the number of edges of type IS, RS or V S will be given respectively by

We define

Then, we update our measures as follows, introducing some notation:

Another event in consideration is a recovering. Here we choose uniformly an infected i and set:

This happens with probability 1/I T − .

The last event is vaccination. The corresponding rate is π t N S t . We remark the strong dependence on the degree of the individual, because is more probable that a higher connectivity node to be vaccinated first. More precisely, the probability that the new vaccinated has degree k is

. Once we draw the vaccinated individual, and supposing her degree is k, we update the measures as follows:

Now we introduce three Poisson Point Measures that will be very useful to describe the M F (N 0 )-valued stochastic process (µ t ) t≥0 . For a similar point of view, see [20] or [5] .

The first one will provide us the possible instant in which an infection occur. We define  dN 1 (s, k, θ 1 , j, l, m, θ 2 , u, θ 3 , v, θ 4 , w, θ 5 ) as a product measure on 3 , where ds and dθ are Lebesgue measures and dn are counting measures on N 0 or U, accordingly.

The degree k infected agent will be connected with j infected, l recovered and m vaccinated agents, drawn according u, v and w as we explained above.

We also have dN 2 (s, i) on E 2 = R + ×N a PPM with intensity β for the recovering process. This is, for each atom we have associated a possible recovering time s and the identification number i of the new recovered.

The last PPM, dN 3 (s, k, θ 1 , j, l, m, θ 2 , u, θ 3 , v, θ 4 , w, θ 5 ) is defined in R + × E 3 where E 3 = E 1 and it is very similar to the first one. It assign a mass to each possible time s where a degree k vaccinated agent is connected with j infected, l recovered and m vaccinated agents, drawn according u, v and w.

In all the cases, the auxiliary variables θ are useful to take in account the rates in this integral representation.

In order to simplify notation we will not write the dependency on the variables, and consider the following indicator functions to represent the rates:

Now it is clear the evolution of the measures according with the events that may occur, and we are ready to write an integral form for this evolution in terms of the Poisson Point Measures, for example for the second coordinate

Doing the same for the four coordinates, we can write the system of Stochastic Differential Equations:

Proposition 4. Given µ 0 = (µ S 0 , µ IS 0 , µ RS 0 , µ V S 0 ) and N 1 , N 2 , N 3 there exists a unique strong solution to the system (26) in the Skorokhod space D(R + , (M F (N 0 )) 4 ).

Proof. First note that all the measures are dominated by the expectation of µ S 0 +µ IS 0 +µ RS 0 + µ V S 0 and the supports are bounded on the positive integers. The proof can be completed in the same way as in [21] .

Inspired in the techniques developed in [5] and [20] we write a renormalization of the system when the number of individuals is n and the number of edges is proportional to n. We observe that the intensity of the jump process has the same order, and deduce the scaling for the fluid limit renormalization. We prove the convergence of the solution of the finite case system of equations to the solution of (26) in the weak sense of the Skorokhod space [22] .

We consider four sequences of measures indexed by n ∈ N, (µ n,S ), (µ n,IS ), (µ n,RS ) and (µ n,V S ) satisfying the system of equations (26) for each n ∈ N with initial conditions µ n,S 0 , µ n,IS 0 , µ n,RS 0 and µ n,V S 0 . We associate S n t , I n t , R n t and V n t the number of individuals in each state at time t and denote S t , I t , R t , V t the sets of the nodes susceptible, infected, recovered and vaccinated, respectively.

We take the scaling µ We assume that the sequences of initial conditions converge weakly in M F (N 0 ) to µ S 0 , µ IS 0 , µ RS 0 and µ V S 0 when n goes to infinity. We obtain the renormalized system

Let us define

Proposition 2. For all f ∈ B b (N) and all t ≥ 0 we have the following decomposition

where the finite variation is given by

and the associated martingale is square integrable with quadratic variation,

Proof. (Sketch) We first calculate the infinitesimal generator L of our process, and we write the Levy's martingale with φ = µ, f and φ 2 . Then we apply the integration by parts formula [23] , and identifying the martingales in the expression, we rearrange the terms in order to get the quadratic variation. For a detailed proof see [20] .

Our fluid limit result may be proved in much the same way as proof of the main theorem of [5] but we add it for completeness.

Proof of Theorem 1. In order to prove (ii), since lim ε →0 t ε = ∞, is enough to prove the result in D([0, t ε ], M 4 0,A ) for ε sufficiently small. From now on, we take 0 < ε < ε < µ IS 0 , χ .

Step 1: Tightness of the renormalization. Take (µ (n) ) n∈N , t ∈ R >0 and n ∈ N. By assumptions, we have:

This implies that the sequence µ (n) t is tight for each t. By the criterion of convergence of measure valued processes proposed by Roelly [24] we have to prove that, for each test function

We present here the calculations only for µ (n),IS , f because the others are similar or simpler. Since we have a semimartingale decomposition, applying the Rebolledo criterion for weak convergence of sequences of semimartingales, we have to prove that both the finite variation part, and the quadratic variation satisfy the Aldous criterion. We want to prove that, for all θ > 0 and η > 0 there exist n 0 ∈ N and δ > 0 such that for all n > n 0 and for all stopping times S n and T n with S n < T n < S n + δ we have

For the finite variation condition (30), we take the following bound:

Since j+l+m≤k p n s (j, l, m|k − 1)2j is twice the mean number of edges with the infected population conditioned to having degree k, this number is bounded by k, and using the definitions of λ n , π and p we have that: Step 3: µ (n) satisfies asymptotically the deterministic system (2) . Let us remember that, for each f ∈ C b (N), we can write: 

where ∆ n,f ·∧τ n ε vanishes in probability and uniformly in t over compact time intervals. We can take bounds in a similar way as in Step 1 in order to get that:

which implies the sequence (M (n),IS,f t ) n∈N vanishes in probability and in L 2 , and therefore in L 1 by Cauchy-Schwartz.

On the other hand, the finite variation part can be split in two: one considering the simple edges between the newly infected node and the infected population, and a second part regarding multiple edges, that we know is expected to vanish as the size of the population grows. Formally, A 

as long as s ≤ τ n ε and n ≥ 1/ε. Additionally, for all the possible draws u ∈ U(j, µ n,IS s ) we have

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Viscosity solutions of hamilton-jacobi equations and optimal control problems

Contrôle optimal: théorie & applications

Convergence of the forward-backward sweep method in optimal control

A microscopic probabilistic description of a locally regulated population and macroscopic approximations

Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques

The theory of stochastic processes I

Continuous martingales and Brownian motion

A criterion of convergence of measure-valued processes: application to measure branching processes

Convergence of probability measures

and applying Markov in the same way, we have the condition for the martingale part (31), so we are in the hypothesis of Aldous-Rebolledo criterion. Therefore, we have proved tightness in D(R + , M 4 0,4 ). 20Now is time to prove the uniqueness of the solution. Before that, observe that, by Step 1 and Prohorov's theorem, the laws of µ (n) for n ∈ N are a family of bounded measures, a precompact set in D(R + , M 4 0,4 ). Hence, also are the laws of the stopped processes (µ (n) ·∧τ n ε ) n∈N . Let µ be a limit point in C(R + , M 4 0,4 ) of the sequence of stopped processes and let (µ (n) ) n∈N be a subsequence that converges to µ, denoted with only one over-script to simplify notation. Since the limit is continuous, the convergence is uniform over compact sets of the positive reals.Define, for all t ∈ R + and f ∈ C b (N) the applications such that (2) can be read asStep 2: Uniqueness of the solution in C(R + , M 0,4 × M 0+,4 × M 0,4 × M 0,4 ). The second step consists in proving the limit values are the unique solution of (2). The strategy will be to prove that the total measure and the first and second moments of two solutions are equals and then prove that the generating functions of those measures satisfies a partial differential equation that admits an unique solution in a weak sense.Due to extension by regularity, is enough to prove the uniqueness in C([0, T ], M 0,4 × M ε,4 × M 0,4 × M 0,4 ) for all ε, T > 0.Take µ i = (µ S,i , µ IS,i , µ RS,i , µ V S,i ) for i = 1, 2 two solutions of (2) in this space with the same initial condition and defineAnalogously, a similar bound holds for |p I,1 t − p I,2 t |. Since µ i are solutions of (2), we have, for j = 0, ..., 3 and taking f = χ j

One can reproduce similar computations for the other quantities and we get Υ t ≤ C(r, β, ν, A, ε) t 0 Υ s ds that is, Υ satisfies a Gronwall type inequality which implies that is identically 0 for all t ≤ T . Then, for all t < T and for j = 1, 2, 3, we have . From the first equation in (2) and the regularity of the solutions, we have almost sure uniqueness for µ S .It remains to prove the uniqueness for the other 3 measures. The method that we will use to prove µ IS,1 = µ IS,2 can be used for the rest.We consider the generating functionsfor any t ∈ R + , i = 1, 2 and η ∈ [0, 1). Let us defineUsing f (k) = η k in the second equation of (2) and after some basic computations we getNow, H(t, η) is continuously differentiable with respect to time and it is well defined and bounded in [0, T ]; and K t is piecewise continuous in L 1 and also it is well defined and bounded on [0, T ]. Further, H and K do not depend on the solution we choose, because we already have µ S,1 = µ S,2 and p I,In view of the regularity of H and K it is known that this equation admits only one solution in a weak sense (see last section in [11] ), hence G 1 and applying both inequalities, for n ≥ 1/ε we get(42) This last expression tends to zero because of the weak convergence of µfor all s ≥ 0 and n ∈ N. The next task is to prove that B (n),IS,f ·∧τ n ε is similar in some way to Ψ IS,f ·∧τ n ε (µ (n) ). For this, we realize thatwhere τ j f (k) := f (k − j) for all f : N → R and ∀k ∈ N. Now we introduce some notation for the proportions of edges the newly infected agent has, discarding the edge involved in the infection process. It is important here to make a difference between the term that comes from the infection from the one that comes from the PPM modelling the vaccination process, because in this case we do not assume a priori that 25 there is at least one infected neighbor. We define, for each t > 0 and n ∈ N 0 ,Let us remember thatIn the case of the infection process, we also define, for all the j, l, m such that j + l + m ≤ k − 1, and for all n ∈ N,and for the vaccinations,the probabilities of the multinomial variables counting the quantities of each types of neighbors that will has the newly infected or vaccinated, respectively. We can write |Bs− q n j,l,m,s ds.(47) Thus, if we consider the differences α n t (k) = j+l+m+1≤k |p n t (j, l, m | k − 1) −p n t (j, l, m | k − 1)| and β n t (k) = j+l+m≤k |q n t (j, l, m | k) −q n t (j, l, m | k)|, we can bound:(48) Since the multinomial term is a good approximation of the multivariate hypergeometric as n goes to infinity, the last expression tends to zero due to dominated convergence.

On the other hand,Putting all the bounds together, we can conclude that µ (n),IS , f converges in probability uniformly over compact intervals.Step 4: The limit satisfies the deterministic system (2) We are considering the sequence (µ (n) ·∧τ n ε ) n∈N and we already proved that its limit in the closed set M 4 0,A is µ, we want to prove the same for the nonstopped sequence. According to the Skorokhod representation theorem there exists a subsequence on the same probability space of µ whose marginal probability distributions are the same as those of the original sequence such that µ is the almost sure limit. With an abuse of notation, we will denote (µ (n) ·∧τ n ε ) n∈N this subsequence. The mappingsAccording to lemma (5) we have that, for p ≤ 5, Φ p : D(R + , M ε,A ) → D(R + , R) which assigns ν · → ν · , χ p is continuous.Using this, and that the quotient (X 1 · , X 2 · ) → X 1 · X 2 · from C(R + , R) × C(R + , R * ) to C(R + , R) is continuous, we deduce the continuity of ν · → ν 1 · ,χ ν 2 · ,χ from C(R + , M 0,A × M ε,A × M 0,A × M 0,A ) in C(R + , R). The same argument holds for ν · → is greater than or equal to ε almost surely.

Let us define t ε = inf{t ∈ R + : N IS t ≤ ε }. We do not know this number to be deterministic, but we can say that:Then, applying Fatou's Lemma,Therefore, splitting in the following way:we have, from the bounds and estimations we made in Step 3, that Ψ IS,f ·∧τ n ε ∧T (µ (n) ) is bounded for the fourth moment of µ (n) . Since µ (n) 0 → µ 0 and using (50), the first term in (51) converges in L 1 and in probability to zero. On the other hand, the continuity of. So, this convergence and (50) implies that, the second term converges to Ψ IS,fStep 3 again, and the estimations done in it, we can conclude this sequence also converges in probability to zero. Therefore, we have that µ IS is a solution to the system (2) on the interval [0, t ε ∧ T ].If either µ RS 0 , χ > 0 or µ V S 0 , χ > 0, then we could apply similar techniques with both. If not, the result can be immediately deduced because for all t ∈ [0, t ε ∧ T ], µ (n),IS t , χ > ε and the terms p n t (j, l, m | k − 1) and q n t (j, l, m | k) are negligible when l or m are positives. So, µ is almost surely the unique continuous solution of the deterministic system (2) in [0, t ε ∧T ], which implies t ε = t ε and the convergence in probability of (µ (n) ·∧τ n ε ) n∈N to µ holds, uniformly on the interval [0, t ε ], due to the continuity of µ.In order to prove the convergence in the Skorokhod space, for η > 0, we write:(52) Using the continuity of Ψ f and the uniform convergence in probability that we have proved, the first term in the last expression converges to zero. In order to show that the second term vanish, we can reproduce the bounds taken in Step 2 of this proof and apply Doob's inequality. Finally, since P (τ n ε > T ∧ t ε → 1 we have that the three terms goes to zero. Hence, due to uniqueness proved in Step 2, the original sequence (µ (n) ) n∈N converges.

Step 5: The convergence of the other measures What we have done for the infectedsusceptible connectivity measure can be also done for the recovered and vaccinated measures in much the same way. For the susceptible connectivity measure, one can reason in the following way. If we consider the renormalized equation 27 and we take limit in n, the sequence (µ (n),S ) n∈N converges in D(R + , M 0,A ) to the solution to the transport equationthat can be solved as a function of p I and π, for any test function f ∈ C 0,1 b (N × R + , R) with bounded derivative respect time variable. If we take f (k, s) = ϕ(k)e − t−s 0 rkp I u +πu(k)du we obtain µ S t , ϕ = k∈N ϕ(k)α k t µ S 0 (k) as the first equation of (2) establishes. The proof is finished.Lemma 5. For any p ≤ 5, the map Φ p : D(R + , M ε,A ) → D(R + , R) that assigns Φ(ν . ) → ν . , χ p is continuous.Proof. The proof can be obtained by following the steps of Lemma 1-5 in the appendix of [5] .